Braiding link cobordisms and non-ribbon surfaces

Hughes, Mark C¹

¹Department of Mathematics, Brigham Young University, 312 TMCB, Provo, UT 84602, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3707-3729

Résumé

We define the notion of a braided link cobordism in S3 × [0,1], which generalizes Viro’s closed surface braids in ℝ4. We prove that any properly embedded oriented surface W ⊂ S3 × [0,1] is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when ∂W already consists of closed braids. These surfaces are closely related to another notion of surface braiding in D2 × D2, called braided surfaces with caps, which are a generalization of Rudolph’s braided surfaces. We mention several applications of braided surfaces with caps, including using them to apply algebraic techniques from braid groups to studying surfaces in 4–space, as well as constructing singular fibrations on smooth 4–manifolds from a given handle decomposition.

Détail

DOI : 10.2140/agt.2015.15.3707

Classification : 57M12, 57M25, 57R52
Keywords: braids, links, knot cobordisms

Bibliographie

[1] S Akbulut, Ç Karakurt, Every $4$–manifold is BLF, J. Gökova Geom. Topol. GGT 2 (2008) 83

[2] J Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923) 93

[3] D Auroux, S K Donaldson, L Katzarkov, Singular Lefschetz pencils, Geom. Topol. 9 (2005) 1043

[4] R İ Baykur, Existence of broken Lefschetz fibrations, Int. Math. Res. Not. 2008 (2008)

[5] R İ Baykur, Broken Lefschetz fibrations and smooth structures on $4$–manifolds, from: "Proceedings of the Freedman Fest" (editors R Kirby, V Krushkal, Z Wang), Geom. Topol. Monogr. 18 (2012) 9

[6] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, de Gruyter (1985)

[7] S Carter, S Kamada, M Saito, Surfaces in $4$–space, Encyclopaedia of Mathematical Sciences 142, Springer, Berlin (2004)

[8] S K Donaldson, Lefschetz fibrations in symplectic geometry, from: "Proceedings of the International Congress of Mathematicians, Extra Vol. II" (1998) 309

[9] P Freyd, D Yetter, J Hoste, W B R Lickorish, K Millett, A Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985) 239

[10] D T Gay, R Kirby, Constructing Lefschetz-type fibrations on four-manifolds, Geom. Topol. 11 (2007) 2075

[11] R E Gompf, A I Stipsicz, $4$–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)

[12] M Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004) 1211

[13] V F R Jones, A polynomial invariant for knots via von Neumann algebras, from: "Fields medallists' lectures" (editors M Atiyah, D Iagolnitzer), World Sci. Ser. $20^{\mathrm{th}}$ Century Math. 5, World Sci. Publ. (1997) 448

[14] S Kamada, $2$–dimensional braids and chart descriptions, from: "Topics in knot theory" (editor M E Bozhüyük), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer (1993) 277

[15] S Kamada, Alexander's and Markov's theorems in dimension four, Bull. Amer. Math. Soc. 31 (1994) 64

[16] S Kamada, A characterization of groups of closed orientable surfaces in $4$–space, Topology 33 (1994) 113

[17] S Kamada, On braid monodromies of non-simple braided surfaces, Math. Proc. Cambridge Philos. Soc. 120 (1996) 237

[18] S Kamada, Arrangement of Markov moves for $2$–dimensional braids, from: "Low-dimensional topology" (editor H Nencka), Contemp. Math. 233, Amer. Math. Soc. (1999) 197

[19] S Kamada, Braid and knot theory in dimension four, Mathematical Surveys and Monographs 95, Amer. Math. Soc. (2002)

[20] M Khovanov, L Rozansky, Matrix factorizations and link homology, II, Geom. Topol. 12 (2008) 1387

[21] Y Lekili, Wrinkled fibrations on near-symplectic manifolds, Geom. Topol. 13 (2009) 277

[22] A Loi, R Piergallini, Compact Stein surfaces with boundary as branched covers of $B^4$, Invent. Math. 143 (2001) 325

[23] H R Morton, An irreducible $4$–string braid with unknotted closure, Math. Proc. Cambridge Philos. Soc. 93 (1983) 259

[24] H R Morton, Threading knot diagrams, Math. Proc. Cambridge Philos. Soc. 99 (1986) 247

[25] L Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 (1983) 1

[26] L Rudolph, Special positions for surfaces bounded by closed braids, Rev. Mat. Iberoamericana 1 (1985) 93

[27] L Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. 29 (1993) 51

[28] L Rudolph, Knot theory of complex plane curves, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier (2005) 349

[29] O Y Viro, lecture (1999)